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प्रश्न
Identify the type of conic and find centre, foci, vertices, and directrices of the following:
9x2 – y2 – 36x – 6y + 18 = 0
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उत्तर
9x2 – 36x – y2 – 6y = – 18
9(x2 – 4x) – (y2 + 6y) = – 18
9(x – 2)2 – (y + 3)2 = – 18 + 36 – 9
9(x – 2)2 – (y + 3)2 = 9
`(x - 2)^2/1 - (y + 3)^2/9` = 1
It is hyperbola.
The transverse axis is parallel to x axis.
a2 = 1, b2 = 9
a = 1, b = 3
c2 = a2 + b2
= 1 + 9
= 10
c = `sqrt(10)`
ae = `sqrt(10)`
e = `sqrt(10)`
Centre (h, k) = (2, – 3)
Vertices (h ± a, k) = (2 ± 1, – 3)
= (2 + 1, –3) and (2 – 1, – 3)
= (3, – 3) and (1, – 3)
Foci (h ± c, k) = `(2 +- sqrt(10), -3)`
= `(2 + sqrt(10), - 3)` and `(2 - sqrt(10), -3)`
Directrix x = `+- "a"/"e" + "h"`
= `+- 1/sqrt(10) + 2` and x = `- 1/sqrt(10) + 2`
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